I don't know anything about being a landlord or homeowner, so I can't advise you about the CA house.

An emergency fund comes after and a little concurrent with accruing your down payment. This is 6-12 months of expenses including the new mortgage payment. I'd say, once you have 3 mo worth saved up, you can go ahead with the house purchase.

It's hard to say how much you should save without knowing your expenses, interest rate on your debt, other assets, etc., but people here aim for 50% of their pay. If your goal is to be Financially Independent/Retire Early (FIRE), too, then 50%. You might consider posting a case study for this community to review.

It looks like you put 10% into a savings account, 10% into your 401K, and you get 6% from your employer. That's 26% for you. You'd need another 24%, plus half your spouse's pay as well, to make it to 50%.